Answers to all your questions:
1. HomeDJ has 2 modes:
Streaming - When Cache is disabled (Settings->Cache) HomeDJ will not store song on the device. It will stream directly from your Home Computer.
Cached Streaming - When Cache is enabled, HomeDJ will start downloading song from Home Computer and saving this song on SD Card of the device. Playback of this song starts as soon as there at least 10 seconds of the song is cached. It does not wait for the whole song to be downloaded before it starts playback. Usually playback start up time is around 1-5 seconds.

Because Streaming mode does not store data on the device, song will be streamed over the Internet every time it plays utilizing Mobile Data plan data, or WiFi. Please note that this mode might cause high data usage.
Cached Streaming will store the song locally on the device and next time you want to play it, it will not be downloaded. This mode helps to minimize data usage as well as preventing player not to stop if you are driving through the bad coverage zones. Cached Streaming has an option to cache ahead up to 15 songs. By default limit on how many songs can be cached is 60% of available storage. If HomeDJ needs more SD Card space to download another song, it will delete oldest songs it downloaded in the past.

2. There is no browser playback support it, but I am working on it. It will be available in a couple of month.

3. Folder scanning - During the scan, HomeDJ is analyzing songs and builds database of fingerprints of each song. Fingerprint is used for song recognition and for MIX creation. MIX is a dynamic playlist, which grows while playing. Based on fingerprint, HomeDJ finds similar song to the one which started that mix to be added to that MIX.
Once scanning is complete, it will not re-scan those songs again. If you want to add more songs to your library, you have to start Folder Scan again. But this time it will only scan just added songs and not the whole library.
If you changed metadata of the song (Title, Artist, Album, Genre, File Name or File location) you have to start folder scan again and HomeDJ will re-scan only changed songs.
Re-scan of just added or changed songs will be much quicker.

There are 3 scan modes:
1. Fast - will only read songs Tags and will try to find fingerprint from central library of fingerprints. If not found, it will mark this song as not Mixable, but you will be able to play it.
2. Normal - will try to recognize songs by song Tag, and songs which were not recognized will be analyzed and fingerprint created for these songs.
3. Full - will try to recognize songs by Tags, but will analyze each song anyway to create cleaner fingerprint. This more is the longest, but because central library of fingerprints now has about 4 Million fingerprints, it is much quicker than it was year or two ago.